Theorem nrexdv 2550

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrexdv.1	|- ( ( ph /\ x e. A ) -> -. ps )

Assertion

Ref	Expression
nrexdv	|- ( ph -> -. E. x e. A ps )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 |- ( ( ph /\ x e. A ) -> -. ps )
2	1	ralrimiva 2530	. 2 |- ( ph -> A. x e. A -. ps )
3		ralnex 2445	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	sylib 121	1 |- ( ph -> -. E. x e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2128   A.wral 2435   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-ie2 1474  ax-4 1490  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-ral 2440  df-rex 2441

This theorem is referenced by:  ltpopr  7515  cauappcvgprlemladdru  7576  cauappcvgprlemladdrl  7577  caucvgprlemladdrl  7598  caucvgprprlemaddq  7628  dvdsle  11736